Can having a broken nose cause post nasal drip? If so is that normal or does it mean you have a deviated septum?

Post nasal drips. A septal deviation with rhinitis can cause post nasal drip, similarly a post nasal drip can occur after acute injuries like fractures though in the latter case it is more likely to be bloody. In either case, it would be best to be seen by your ENT doctor or your plastic surgeon to get appropriate care.
Nasal fracture. You may have a post nasal drip with a broken nose as well as a deviated septum treatment cosists of antihistamines or possible surgery on the nose.
Post nasal drip. Post nasal drip is not caused by a deviated septum. Rather it is caused by allergic rhinitis. You can have your nose evaluated by a board certified otolaryngologist.